
which, corresponds to Pryce, Newton, and Wigner gauge for spin supplementarity condition (SSC) given by \(S_{(a)\mu \nu } (u_{(a)}^\nu + \sqrt{u_{(a)}^2} \delta ^{\nu 0})\approx 0\) .
In the above action, the center of mass of the compact object is modeled by the position of the point particle \(x_{(a)}^\mu \)  and the spin of the compact object is modeled by \(S_{(a)\mu \nu }\) . The \(\mathcal {L}_{(a)\text{SI}}\)  is given by equation (4.16) of [1]} which corresponds to the spin-induced non minimal couplings that does not contribute up to the N\(^3\) LO spin-obit sector at 4.5PN.
Here \(u_{(a)}^\mu =dx_{(a)}^\mu /d\tau \)  is the four-velocity and \(u_{(a)}^2=g_{\mu \nu }u_{(a)}^\mu u_{(a)}^\nu \) , whereas the proper time \(\tau \)  is related to the coordinate time \(t\)  by \(d\tau =c~ dt\) .
